New proposals for modelling and solving the problem of covering solids using spheres of different radii

¹ Departamento de Informática, Centro Federal de Educação Tecnológica Celso Suckow da Fonseca, Brazil
² Departamento de Estatística, Universidade Federal da Paraíba, Brazil
³ Departamento de Matemática, Universidade Federal Rural do Rio de Janeiro, Brazil
⁴ Programa de Engenharia de Sistemas e Computação, Universidade Federal do Rio de Janeiro, Brazil
⁵ Laboratoire de Mathématiques d’Avignon, Université d’Avignon et des Pays de Vaucluse, France

^* Corresponding author: pehgonzalez@gmail.com; pegonzalez@eic.cefet-rj.br

Received: 27 September 2017
Accepted: 7 April 2019

Abstract

Given a solid T, represented by a compact set in ℝ³, the aim of this work is to find a covering of T by a finite set of spheres of different radii. Some level of intersection between the spheres is necessary to cover the solid. Moreover, the volume occupied by the spheres on the outside of T is limited. This problem has an application in the planning of a radio-surgery treatment known by Gamma Knife and can be formulated as a non-convex optimization problem with quadratic constraints and linear objective function. In this work, two new linear mathematical formulations with binary variables and a hybrid method are proposed. The hybrid method combines heuristic, data mining and an exact method. Computational results show that the proposed methods outperform the ones presented in the literature.